Related papers: L-spaces, taut foliations and the Whitehead link
Motivated by the $L$-space conjecture, we prove left-orderability of certain Dehn fillings on integral homology solid tori with techniques first appearing in the work of Culler-Dunfield. First, we use the author's previous results to…
A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c…
We prove that the link of a complex normal surface singularity is an L--space if and only if the singularity is rational. This via a recent result of Hanselman, J. Rasmussen, S. D. Rasmussen and Watson (proving the conjecture of Boyer,…
This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In…
We prove that if a knot $K$ has a particular type of diagram then all non-trivial surgeries on $K$ contain a coorientable taut foliation. Knots admitting such diagrams include many two-bridge knots, many pretzel knots, many Montesinos knots…
We classify all contact structures with contact surgery number one on the Brieskorn sphere Sigma(2,3,11) with both orientations. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds…
We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r <…
Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres obtained by gluing compact manifolds along torus…
We introduce the notion of a strong L-space, a closed, oriented rational homology 3-sphere whose Heegaard Floer homology can be determined at the chain level. We prove that the fundamental group of a strong L-space is not left-orderable.…
An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link $K_{rm,rn}$ is an L-space link if and only if K is an L-space knot and…
An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double…
We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincar\'e homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not…
We describe necessary and sufficient conditions for a knot in an L-space to have an L-space homology sphere surgery. We use these conditions to reformulate a conjecture of Berge about which knots in S^3 admit lens space surgeries.
Much work has been done recently towards trying to understand the topological significance of being an L-space. Building on work of Rasmussen and Rasmussen, we give a topological characterisation of Floer simple manifolds such that all…
We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^3. Our main tool is a careful use of the Aarhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N,…
Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer…
The Cyclic Surgery Theorem and Moser's work on surgeries on torus knots imply that for any non-trivial knot in $S^3$, there are at most two integer surgeries that produce a lens space. This paper investigates how many positive integer…
An $L$-space link is a link in $S^3$ on which all large surgeries are $L$-spaces. In this paper, we initiate a general study of the definitions, properties, and examples of $L$-space links. In particular, we find many hyperbolic $L$-space…
The $L$-space conjecture asserts the equivalence, for prime $3$-manifolds, of three properties: not being an $L$-space ($NLS$), having a left-orderable fundamental group ($LO$), and admitting a co-orientable taut foliation ($CTF$). In this…
We provide related Dehn surgery descriptions for rational homology spheres and a class of their regular finite cyclic covering spaces. As an application, we use the surgery descriptions to relate the Casson invariants of the covering spaces…